Download 4 CHAPTER SUMMARY

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4 CHAPTER SUMMARY
What did you learn?
EXAMPLE(S)
REVIEW
EXERCISES
1
1–6
3–4
7–10
2, 5, 6
11–14
7
15–16
1
17–20
2, 4–6
21–24
3, 7, 8
25–28
1
29, 30
2, 3
31–33
Section 4.1
쐽
How to distinguish between discrete random variables and continuous
random variables
쐽
How to determine if a distribution is a probability distribution
쐽
How to construct a discrete probability distribution and its graph and find the
mean, variance, and standard deviation of a discrete probability distribution
m = gxP1x2
Mean of a discrete random variable
s = g1x - m2 P1x2
2
2
s = 2s = 2g1x - m2 P1x2
2
쐽
2
Variance of a discrete random variable
Standard deviation of a discrete
random variable
How to find the expected value of a discrete probability distribution
Section 4.2
쐽
How to determine if a probability experiment is a binomial experiment
쐽
How to find binomial probabilities using the binomial probability formula,
a binomial probability table, and technology
P1x2 = nCx p xq n - x =
쐽
n!
p xq n - x
1n - x2!x!
Binomial probability formula
How to construct a binomial distribution and its graph and find the mean,
variance, and standard deviation of a binomial probability distribution
m = np
Mean of a binomial distribution
2
s = npq
Variance of a binomial distribution
s = 1npq
Standard deviation of a binomial distribution
Section 4.3
쐽
How to find probabilities using the geometric distribution
P1x2 = pq
쐽
x-1
Probability that the first success will occur on trial number x
How to find probabilities using the Poisson distribution
x -m
P1x2 =
me
x!
Probability of exactly x occurrences in an interval
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R E V I E W E XE RCIS E S
4
227
REVIEW EXERCISES
쏋 SECTION 4.1
In Exercises 1 and 2, decide whether the graph represents a discrete random
variable or a continuous random variable. Explain your reasoning.
1. The number of hours spent
sleeping each day
0
4
8
12
16
20
2. The number of fish caught during
a fishing tournament
24
0
2
4
6
8
10
In Exercises 3–6, decide whether the random variable x is discrete or continuous.
3. Let x represent the number of pumps in use at a gas station.
4. Let x represent the weight of a truck at a weigh station.
5. Let x represent the amount of carbon dioxide emitted from a car’s tailpipe
each day.
6. Let x represent the number of people that activate a metal detector at an
airport each hour.
In Exercises 7–10, decide whether the distribution is a probability distribution. If it
is not, identify the property that is not satisfied.
7. The daily limit for catching bass at a lake is four. The random variable x
represents the number of fish caught in a day.
x
P(x)
0
1
2
3
4
0.36
0.23
0.08
0.14
0.29
8. The random variable x represents the number of tickets a police officer
writes out each shift.
x
P(x)
0
1
2
3
4
5
0.09
0.23
0.29
0.16
0.21
0.02
9. A greeting card shop keeps records of customers’ buying habits. The random
variable x represents the number of cards sold to an individual customer in
a shopping visit.
x
P(x)
1
2
3
4
5
6
7
0.68
0.14
0.08
0.05
0.02
0.02
0.01
10. The random variable x represents the number of classes in which a student is
enrolled in a given semester at a university.
x
1
2
3
4
5
6
7
8
P(x)
1
80
2
75
1
10
12
25
27
20
1
5
2
25
1
120
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In Exercises 11–14,
(a) use the frequency distribution table to construct a probability distribution,
(b) graph the probability distribution using a histogram and describe its shape,
(c) find the mean, variance, and standard deviation of the probability distribution, and
(d) interpret the results in the context of the real-life situation.
11. The number of pages in a section
from 10 statistics texts
Pages
12. The number of hits per game played
by a baseball player during a recent
season
Sections
Hits
Games
2
3
0
29
3
12
1
62
4
72
2
33
5
115
3
12
6
169
4
3
7
120
5
1
8
83
9
48
10
22
11
6
13. The distribution of the number of
cellular phones per household in
a small town is given.
Cellphones
Families
0
5
1
35
2
68
3
73
4
42
5
19
6
8
14. A television station sells advertising
in 15-, 30-, 60-, 90-, and 120-second
blocks. The distribution of sales for
one 24-hour day is given.
Length
(in seconds)
Number
15
76
30
445
60
30
90
3
120
12
In Exercises 15 and 16, find the expected value of the random variable.
15. A person has shares of eight different stocks. The random variable x
represents the number of stocks showing a loss on a selected day.
0
1
2
3
4
5
6
7
8
0.02
0.11
0.18
0.32
0.15
0.09
0.05
0.05
0.03
x
P(x)
16. A local pub has a chicken wing special on Tuesdays. The pub owners purchase
wings in cases of 300. The random variable x represents the number of cases
used during the special.
x
1
2
3
4
P(x)
1
9
1
3
1
2
1
18
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쏋 SECTION 4.2
In Exercises 17 and 18, use the following information.
A probability experiment has n independent trials. Each trial has three possible
outcomes: A, B, and C. For each trial, P1A2 = 0.30, P1B2 = 0.50, and
P1C2 = 0.20. There are 20 trials.
17. Can a binomial experiment be used to find the probability of 6 outcomes of
A, 10 outcomes of B, and 4 outcomes of C? Explain your reasoning.
18. Can a binomial experiment be used to find the probability of 4 outcomes
of C and 16 outcomes that are not C? Explain your reasoning. What is the
probability of success for each trial?
In Exercises 19 and 20, decide whether the experiment is a binomial experiment. If
it is not, identify the property that is not satisfied. If it is, list the values of n, p, and
q and the values that x can assume.
19. Bags of plain M&M’s contain 24% blue candies. One candy is selected from
each of 12 bags. The random variable represents the number of blue candies
selected. (Source: Mars, Incorporated)
20. A fair coin is tossed repeatedly until 15 heads are obtained. The random
variable x counts the number of tosses.
In Exercises 21–24, find the indicated probabilities.
21. One in four adults is currently on a diet. You randomly select eight adults
and ask them if they are currently on a diet. Find the probability that the
number who say they are currently on a diet is (a) exactly three, (b) at least
three, and (c) more than three. (Source: Wirthlin Worldwide)
22. One in four people in the United States owns individual stocks. You
randomly select 12 people and ask them if they own individual stocks. Find
the probability that the number who say they own individual stocks is
(a) exactly two, (b) at least two, and (c) more than two. (Source: Pew Research
Center)
23. Forty-three percent of businesses in the United States require a doctor’s
note when an employee takes sick time. You randomly select nine businesses
and ask each if it requires a doctor’s note when an employee takes sick time.
Find the probability that the number who say they require a doctor’s note is
(a) exactly five, (b) at least five, and (c) more than five. (Source: Harvard
School of Public Health)
24. In a typical day, 31% of people in the United States with Internet access
go online to get news. You randomly select five people in the United States
with Internet access and ask them if they go online to get news. Find the
probability that the number who say they go online to get news is (a) exactly
two, (b) at least two, and (c) more than two. (Source: Pew Research Center)
In Exercises 25–28,
(a) construct a binomial distribution,
(b) graph the binomial distribution using a histogram and describe its shape,
(c) find the mean, variance, and standard deviation of the binomial distribution
and interpret the results in the context of the real-life situation, and
(d) determine the values of the random variable x that you would consider unusual.
25. Thirty-four percent of women in the United States say their spouses never
help with household chores. You randomly select five U.S. women and ask if
their spouses help with household chores. (Source: Boston Consulting Group)
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26. Sixty-eight percent of families say that their children have an influence on
their vacation destinations. You randomly select six families and ask if their
children have an influence on their vacation destinations. (Source: YPB&R)
27. In a recent year, forty percent of trucks sold by a company had diesel
engines. You randomly select four trucks sold by the company and check if
they have diesel engines.
28. Sixty-three percent of U.S. mothers with school-age children choose fast
food as a dining option for their families one to three times a week. You
randomly select five U.S. mothers with school-age children and ask if they
choose fast food as a dining option for their families one to three times a
week. (Adapted from Market Day)
쏋 SECTION 4.3
In Exercises 29–32, find the indicated probabilities using the geometric distribution
or the Poisson distribution. Then determine if the events are unusual. If convenient,
use a Poisson probability table or technology to find the probabilities.
29. Twenty-two percent of former smokers say they tried to quit four or more
times before they were habit-free. You randomly select 10 former smokers.
Find the probability that the first person who tried to quit four or more times
is (a) the third person selected, (b) the fourth or fifth person selected, and
(c) not one of the first seven people selected. (Source: Porter Novelli Health
Styles)
30. In a recent season, hockey player Sidney Crosby scored 33 goals in 77 games
he played. Assume that his goal production stayed at that level the following
season. What is the probability that he would get his first goal
(a)
(b)
(c)
(d)
in the first game of the season?
in the second game of the season?
in the first or second game of the season?
within the first three games of the season? (Source: ESPN)
31. During a 69-year period, tornadoes killed 6755 people in the United States.
Assume this rate holds true today and is constant throughout the year. Find
the probability that tomorrow
(a)
(b)
(c)
(d)
no one in the U. S. is killed by a tornado,
one person in the U.S. is killed by a tornado,
at most two people in the U.S. are killed by a tornado, and
more than one person in the U.S. is killed by a tornado. (Source: National
Weather Service)
32. It is estimated that sharks kill 10 people each year worldwide. Find the
probability that at least 3 people are killed by sharks this year
(a) assuming that this rate is true,
(b) if the rate is actually 5 people a year, and
(c) if the rate is actually 15 people a year. (Source: International Shark Attack
File)
33. In Exercise 32, describe what happens to the probability of at least three
people being killed by sharks this year as the rate increases and decreases.
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C H AP T E R QU IZ
231
4 CHAPTER QUIZ
Take this quiz as you would take a quiz in class. After you are done, check your
work against the answers given in the back of the book.
1. Decide if the random variable x is discrete or continuous. Explain your
reasoning.
(a) Let x represent the number of lightning strikes that occur in Wyoming
during the month of June.
(b) Let x represent the amount of fuel (in gallons) used by the Space Shuttle
during takeoff.
2. The table lists the number of U.S. mainland hurricane strikes (from 1851 to
2008) for various intensities according to the Saffir-Simpson Hurricane Scale.
Intensity
Number of hurricanes
1
114
2
74
3
76
4
18
5
3
TABLE FOR EXERCISE 2
(Source: National Oceanic and Atmospheric Administration)
(a) Construct a probability distribution of the data.
(b) Graph the discrete probability distribution using a probability histogram.
Then describe its shape.
(c) Find the mean, variance, and standard deviation of the probability
distribution and interpret the results.
(d) Find the probability that a hurricane selected at random for further study
has an intensity of at least four.
3. The success rate of corneal transplant surgery is 85%. The surgery is
performed on six patients. (Adapted from St. Luke’s Cataract & Laser Institute)
(a) Construct a binomial distribution.
(b) Graph the binomial distribution using a probability histogram. Then
describe its shape.
(c) Find the mean, variance, and standard deviation of the probability
distribution and interpret the results.
(d) Find the probability that the surgery is successful for exactly three patients.
Is this an unusual event? Explain.
(e) Find the probability that the surgery is successful for fewer than four
patients. Is this an unusual event? Explain.
4. A newspaper finds that the mean number of typographical errors per page is
five. Find the probability that
(a) exactly five typographical errors will be found on a page,
(b) fewer than five typographical errors will be found on a page, and
(c) no typographical errors will be found on a page.
In Exercises 5 and 6, use the following information. Basketball player Dwight
Howard makes a free throw shot about 60.2% of the time. (Source: ESPN)
5. Find the probability that the first free throw shot Dwight makes is the fourth
shot. Is this an unusual event? Explain.
6. Find the probability that the first free throw shot Dwight makes is the second
or third shot. Is this an unusual event? Explain.
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PUTTING IT ALL TOGETHER
Real Statistics — Real Decisions
The Centers for Disease Control and Prevention (CDC) is required by
law to publish a report on assisted reproductive technologies (ART).
ART includes all fertility treatments in which both the egg and the
sperm are used. These procedures generally involve removing eggs
from a woman’s ovaries, combining them with sperm in the laboratory,
and returning them to the woman’s body or giving them to another
woman.
You are helping to prepare the CDC report and select at random
10 ART cycles for a special review. None of the cycles resulted in a
clinical pregnancy. Your manager feels it is impossible to select at
random 10 ART cycles that did not result in a clinical pregnancy. Use
the information provided at the right and your knowledge of statistics
to determine if your manager is correct.
Results of ART Cycles
Using Fresh Nondonor
Eggs or Embryos
Ectopic pregnancy
0.7%
Clinical
pregnancy
34.9%
No pregnancy
64.3%
(Source: Centers for Disease Control and Prevention)
쏋 EXERCISES
1. How Would You Do It?
(a) How would you determine if your manager’s view is correct,
that it is impossible to select at random 10 ART cycles that did
not result in a clinical pregnancy?
(b) What probability distribution do you think best describes the
situation? Do you think the distribution of the number of
clinical pregnancies is discrete or continuous? Why?
(a) Selecting at random 10 ART cycles among women of age 40,
eight of which resulted in clinical pregnancies.
(b) Selecting at random 10 ART cycles among women of age 41,
none of which resulted in clinical pregnancies.
30
40
41
43
6.9
12.4
42
6.2
19.7
22.6
5
8.6
10
12.4
15
15.4
20
15.0
Pregnancy rate
Live birth rate
25
Percentage
2. Answering the Question
Write an explanation that answers the question, “Is it possible to
select at random 10 ART cycles that did not result in a clinical
pregnancy?” Include in your explanation the appropriate
probability distribution and your calculation of the probability of
no clinical pregnancies in 10 ART cycles.
3. Suspicious Samples?
Which of the following samples would you consider suspicious if
someone told you that the sample was selected at random? Would
you believe that the samples were selected at random? Why or
why not?
Pregnancy and Live Birth Rates for
ART Cycles Among Women
of Age 40 and Older
2.7 3.71.5
44
Age
(Source: Centers for Disease Control and Prevention)
45
and older
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T ECH N OLOG Y
TECHNOLOGY
MINITAB
EXCEL
USING POISSON DISTRIBUTIONS
AS QUEUING MODELS
TI-83/84 PLUS
M I N I TA B
Probability
0.2
Queuing means waiting in line to be served. There are many
examples of queuing in everyday life: waiting at a traffic light, waiting
in line at a grocery checkout counter, waiting for an elevator, holding
for a telephone call, and so on.
Poisson distributions are used to model and predict the number
of people (calls, computer programs, vehicles) arriving at the line. In
the following exercises, you are asked to use Poisson distributions to
analyze the queues at a grocery store checkout counter.
0.1
0
0 2 4 6 8 10 12 14 16 18 20
Number of arrivals per minute
쏋 EXERCISES
In Exercises 1–7, consider a grocery store that can
process a total of four customers at its checkout
counters each minute.
1. Suppose that the mean number of customers
who arrive at the checkout counters each
minute is 4. Create a Poisson distribution with
m = 4 for x = 0 to 20. Compare your results
with the histogram shown at the upper right.
2. MINITAB was used to generate 20 random
numbers with a Poisson distribution for m = 4.
Let the random number represent the number
of arrivals at the checkout counter each minute
for 20 minutes.
3
3
3
5
3
6
3
3
5
4
5
6
6
2
7
2
3
4
6
1
4. Suppose that the mean increases to 5 arrivals
per minute. You can still process only four per
minute. How many would you expect to be
waiting in line after 20 minutes?
5. Simulate the setting in Exercise 4. Do this by
generating a list of 20 random numbers with a
Poisson distribution for m = 5. Then create a
table that shows the number of customers
waiting at the end of 20 minutes.
6. Suppose that the mean number of arrivals
per minute is 5. What is the probability that
10 customers will arrive during the first minute?
7. Suppose that the mean number of arrivals per
minute is 4.
During each of the first four minutes, only
three customers arrived. These customers could
all be processed, so there were no customers
waiting after four minutes.
(a) How many customers were waiting after
5 minutes? 6 minutes? 7 minutes? 8 minutes?
(b) Create a table that shows the number of
customers waiting at the end of 1 through
20 minutes.
3. Generate a list of 20 random numbers with a
Poisson distribution for m = 4. Create a table
that shows the number of customers waiting at
the end of 1 through 20 minutes.
Extended solutions are given in the Technology Supplement.
Technical instruction is provided for MINITAB, Excel, and the TI-83/84 Plus.
233
(a) What is the probability that three, four, or
five customers will arrive during the third
minute?
(b) What is the probability that more than four
customers will arrive during the first
minute?
(c) What is the probability that more than four
customers will arrive during each of the
first four minutes?